Setting Scale Z of object to zero for complete movie?
You could try to go in edit mode, and then select the vertices you don't mind (press C and then use the circle to select them) and then delete them by pressing X and the option "vertices". Otherwise I think that for make the eye moving you need key-frames. If you selected the option LocRotScale every time you reach the key-frame, it set the location (Loc), the rotation (Rot), and the scale (Scale) of the object to the value it had when you have set the key-frames. So basically you have to delete them, set the Z scale of the object to zero, and re-set the key-frames. To delete the key-frames, select the object, go on the frame on which there is the key-frame, and press Alt-I. Hope helps.